Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18)(19,31)(20,26)(21,27)(22,28)(23,29)(24,30)(25,34)(32,47)(33,42)(35,41)(36,43)(37,44)(38,45)(39,46)(40,48);(2,5)(6,8)(17,19,23,34)(18,25,29,31)(20,22,33,24)(21,36,38,35)(26,30,42,28)(27,41,45,43)(32,37,48,39)(40,44,47,46);(9,11,12,16)(10,13,14,15)(17,20,24,19,23,33,22,34)(18,26,30,31,29,42,28,25)(21,36,39,48,38,35,37,32)(27,43,46,40,45,41,44,47);(17,21,22,37,23,38,24,39)(18,27,28,44,29,45,30,46)(19,32,20,35,34,48,33,36)(25,40,42,43,31,47,26,41);(1,3,4,7)(2,6,5,8)(17,22,23,24)(18,28,29,30)(19,33,34,20)(21,37,38,39)(25,26,31,42)(27,44,45,46)(32,36,48,35)(40,41,47,43)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373624);